No row of a Legendre pair of length 153 with prescribed 9-compression has a nontrivial affine symmetry

Let (a, b) be a Legendre pair of length 153 = 9 · 17 whose 9-compressions are δ + 3χ and δ − 3χ, the compressions prescribed by Kotsireas's conjecture for lengths pq². Here χ is the quadratic character modulo 17, with value 0 at zero, and δ is the indicator of zero. We show that no affine bijection i ↦ ui + c of ℤ/153ℤ other than the identity fixes a or b. Centring and the structure of the unit group reduce the question to four multipliers. The multiplier 52, of order three, is excluded by a known obstruction for lengths p²q, which has a Lean proof. For each of the three involutions 35, 118 and 152, exhaustive searches show that no prescribed pair has a row fixed by it; every part of these searches was carried out by two programs with separately written search code. The unit 118 plays the role that 64 plays at length 117, where the prescribed pairs with both rows fixed by 64 form one orbit under the equivalences that preserve the prescription. Whether a prescribed pair of length 153 exists remains open.

This record holds the article, its source archive, and six search packages covering the computations of Sections 4 and 5, each with its verification receipt. The source archive contains the LaTeX source; the Lean sources, with a command that compiles them from a clean state and replays every proof in the Lean kernel and in the independent checker nanoda, and the receipt of a passing replay; and a check script that verifies the packages against the SHA-256 values in search/PACKAGES.json and repeats the finite computations of the written proofs. Each package has a README with the commands that rerun its controls and a sample of its searches. The article and the data are licensed under CC BY 4.0, and the programs and the Lean sources under the MIT License, as LICENSES.md states; vendored third-party libraries keep their own terms.

Use of generative AI. OpenAI's ChatGPT and Codex, together with Anthropic's Claude, were used in developing the mathematics, writing and checking the programs, and preparing the text. The author is responsible for the mathematical statements, computations and references.